CHARACTERIZATION OF THE SPHERICAL STABILIZATION FLOW
REGIME BY TRANSIENT PRESSURE ANALYSIS
Freddy Humberto Escobar1, Alfredo Ghisays-Ruiz2 and Priyank Srivastav3
1Universidad Surcolombiana/CENIGAA, Avenida Pastrana, Neiva, Huila, Colombia
2Universidad del Atlantico, Fac. de Ciencias Básicas, Antigua vía Puerto Colombia, Barranquilla, Atlántico, Colombia 3The University of Oklahoma, Mewborne School of Petroleum and Geological Engineering, E. Boyd St. SEC Rm, Norman, USA
E-Mail: [email protected]
ABSTRACT
There are three typical cases in which a constant-pressure boundary is combined with some other transient periods giving origin to the formation of new flow regimes. Such cases are radial stabilization, linear stabilization and spherical stabilization. The first one when a radial flow regime finds a constant-pressure boundary, the late pressure derivative will display a straight line with a negative unit slope. Once all the boundaries have been felt by the transient wave the pressure derivative will take the classic cascade behavior. The third case takes place in elongated system when the well is near a lateral pressure-constant boundary, then a transient period is expected along the other side of the reservoir. A combination of that with the effect of the constant–pressure boundary leads to the formation of the linear stabilization or parabolic flow regime. The third case corresponds to a limited-entry well completed near a constant-pressure boundary. In that case a -3/2 slope is seen in the pressure derivative plot and no characterization of this has ever been presented in the literature. So, a governing equation for such flow regime is developed and characterization of that is achieved by both conventional analysis and the TDS technique so both vertical and horizontal permeabilities can be estimated. Synthetic examples were run to validate the applicability of the provided equations.
Keywords: spherical stabilization, vertical anisotropy, limited entry, partial penetration, partial completion, formation testing
1. INTRODUCTION
With the advent of newer technologies like wireline formation testing (WFT) tools for formation pressure testing, it is now possible to gain critical information about reservoir just after drilling of well. In many cases as shown by Frimann-Dahl et al. (1998) it has now become possible to replace the expensive drill stem tests which may take days to months with short duration formation pressure tests also called as Mini-Dst’s, although these Mini-Dst’s provide small radius of investigation as compared to normal well tests, nevertheless they can be analyzed using same well test analyses principles as illustrated by Daungkaew et al. (2004). In addition Daungkaew. S et al showed that WFT provides information about localized near wellbore phenomenon which may be masked in convectional well tests. While several probe configurations exists for conducting the test almost all WFT rely on inserting a probe in virgin reservoir and pumping reservoir fluids from probe for creating drawdown, accurate quartz gauges allows measurement of sand face pressure with time. Every WFT pressure data is expected to have spherical flow, Stewart and Wittmann (1979) introduced analytical solutions for analysis of spherical flow regime in WFT. If we want to extract maximum information from these tests, it becomes critically important to properly classify spherical flow and its transition to other regimes. In 1978, Tippie and Abbot showed that Analysis of pressure transient data in bottom water drive with partial
completion displays flow geometry that changes with time from spherical to hemispherical to linear flow depending on distance of perforations or position of probe in case of wireline formation tests with vertical reservoir boundaries. For this type of reservoir setting existing flow regimes does not adequately describe the flow system.
elongated reservoirs. As far as transient rate analysis is concerned, the researches of Escobar, Rojas and Bonilla (2012a) Escobar Rojas and Cantillo (2012b) included the characterization of the parabolic flow regime.
No information was found for the characterization of the spherical stabilization flow regime which takes place in a limited-entry well completed near a constant-pressure boundary. A governing model for limited-entry wells with bottom constant-pressure boundary was presented by Ichara (1981) and a conventional analysis was introduced by Tippie and Abbot (1978) but they did not elaborate on the -3/2 slope.
This paper presents the recognition of a new flow regime for case of spherical flow under partial penetration where main hydrocarbon column is supported by bottom water drive or gas cap drive on top. This spherical partial penetration/Spherical stabilization flow regime have pressure derivative characteristic slope of -3/2 (although, since it is near boundary, it should be better called hemispherical stabilization) in contrast to -1/2 slope that is observed during normal spherical flow. This increase in slope can be envisioned as support provided by constant pressure boundary (gas cap or water drive) and thus development of this flow regime is dependent upon distance of perforation to boundaries as well as permeability anisotropy.
This paper also tries to demonstrate significance of newly found spherical partial penetration flow regime for estimation of permeability anisotropy of a reservoir system. This can provide us with an additional tool for estimation of reservoir permeability anisotropy where we do not have well defined spherical flow present in the pressure derivative diagnostic curves. Both TDS
technique, Tiab (1993), and conventional analysis were implemented for the characterization of such regime. They were tested with synthetic examples.
2. MATHEMATICAL FORMULATION
2.1. Mathematical model
Ichara (1981) presented the solution for pressure behavior of a limited-entry well with a constant-pressure bottom boundary. The dimensionless time, pressure and pressure derivative used in this work are given as:
2 0.0002637
D
t w
kt t
c r
(1)
141.2 D
kh P P
q B
(2)
( * ') * '
141.2 D D
kh t P t P
q B
(3)
The vertical anisotropy or permeability ratio, Iv, is
here defined as:
z v
k I
k
(4)
1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10
t D
t *P
'
D
D
h 300 200 50 100 200
hp 45 40 5 30 20
kz/k 10 1 0.01 0.1 0.001 slope=-3/2
Figure-1. Dimensionless pressure derivative versus time
log-log behavior for several values of reservoir thickness, perforated thickness and vertical/horizontal
permeability ratio
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08
t D
t
*P
'
D
D
1 v
I
0.01 v
I
10 v
I
0 10 20 D
C
2000 1000 100 0
D
C
100 10 0
D
C
slope= -3/2
200 ft 40 ft p
h h
Figure-2. Effect of wellbore storage on the spherical
stabilization flow regime.
And the penetration ratio, b, is defined as:
p h b
h
(5)
that a vertical/horizontal permeability ratio is small. Notice for the case of a smaller permeability ratio, last curve in the right, the spherical stabilization is seen for a reservoir thickness of 200 ft. Penetration ratios higher than 40 % avoids development of this flow regime. Notice that for low permeability contrast the radial flow is almost seen since the effect of the constant-pressure boundary is retarded. The complete steady-state period is fully developed once the transient wave has reached the no-flow pressure boundary; the thicker the reservoir the later the maximum point is seen. This maximum corresponds to both the presence of the no-flow boundary and the penetration ratio.
1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
3/ 2 max * ' /
0.000575 ( / )
D D w
p
t P h r
h h
Spherical stabilization slope= -3/2
2 2
max
10.9387
w z
D
r k t h k
(k /kz w D )(r /h) t
2
D
D
[
t
*
P
'
](
h
/
h
)
(
h
/
r
)
p
-3
/2
w
1/
2
Figure-3. Unified behavior of the dimensionless pressure
derivative versus time log-log.
Figure-2 is a dimensionless pressure derivative versus dimensionless time log-log plot for several values of dimensionless wellbore storage and anisotropy ratio. Spherical stabilization flow regime is affected more by wellbore storage for reservoir with higher anisotropy ratio. For instances when the anisotropy ratio is 10, meaning vertical permeability is 10 times higher than the horizontal permeability, dimensionless wellbore storage greater than 20 will mask the spherical stabilization flow regime.
However, for the isotropic case the onset is at dimensionless wellbore storage of 100. Finally, when the permeability ratio is 0.01, dimensionless wellbore storage values up to 2000 allow seeing the spherical stabilization flow. Generally for reservoirs containing hydrocarbons due to geological sedimentary deposition horizontal permeability is greater than vertical permeability; hence, anisotropy ratio will be less than 1, therefore we expect to see spherical stabilization flow regime even for higher wellbore storage.
Figure-3 presents a unified pressure derivative curve for different values of reservoir thickness, thickness penetration ratio and vertical/horizontal permeability ratio. All the curves fall into a single one if the dimensionless time is multiplied by the permeability ratio and the pressure derivative is multiplied by the penetration ratio to the power -3/2. This allows developing the following mathematical model by regression analysis:
3/ 2 2
1
*
'
11
w z w
D D D
p
r
k
r
t
P
t
h
h k h
(6)Integrating the above expression leads to:
3/2 2
2
1
33
w z w
D D sps
p
r
k
r
P
t
s
h
h k h
(7)The spherical stabilization skin factor, ssps, is
assumed to be a combination of the mechanical skin factor and the vertical flow.
2.2. TDS technique
This technique was introduced by Tiab (1995) and is based upon finding characteristic feature and points in the pressure and pressure derivative log-log plot. Replacing the dimensionless parameters given by Equations (1) and (3) into Equation (6) and solving for the radial permeability:
3/2 5/2
9417306.154
( *
')
p t
w
sps z sps
h c
q
B r
k
t
P
k t
(8)The pressure derivative (t*P’)sps is read at any
convenient time, tsps, during the spherical stabilization
flow regime. Equation (8) assumes that the value of the vertical permeability is known which may be obtained from a repeat wireline formation test. When the reservoir anisotropy as defined here is very low, see Figure-1, it is possible to see the radial flow regime. In such case, a horizontal line is drawn along the radial flow regime and the pressure derivative corresponding to such line is read and horizontal permeability can be obtained from the following expression developed by Tiab (1993):
70.6 ( * ')r
q B k
h t P
(9)
If this is the case, then, it is better to solve for the vertical permeability from Equation (8).
2/3 5/3 44594.83
( * ')
p t z
sps sps h c
q B
k
k t P t
(10)
2 66694.8
( * ') 3
p t sps
w
z sps sps
h c P
s h r
k t t P
(11)Once horizontal permeability is calculated from Equation (8), the radial pressure derivative can be estimated from Equation (8), so:
70.6 ( * ')r
q B t P
hk
(12)
This value may be drawn as a horizontal line on the pressure derivative plot. This horizontal line takes the value of 0.5 in dimensionless form. Therefore, the intersection of the horizontal line with the spherical stabilization line allows obtaining the vertical permeability. The below equation is derived by setting the left-hand side of Equation (6) to 0.5 and solving for kz:
1/3 2/3
2610.65
p t wz
spsri
h
c r h
k
t
(13)The normal case is that both vertical and horizontal permeabilities are unknown. In such case is when the TDS technique shows its power, capability and practicality. As observed in Figure-3, the coordinates of the maximum point are:
3/ 2
max
*
'
/
0.000575
(
/ )
D D w
p
t
P
h r
h
h
(14) 2 2 max10.9387
w z Dr k
t
h k
(15)When the dimensionless pressure derivative given by Equation (3) is replaced into Equation (14) and dimensionless time given by Equation (1) is plugged in Equation (15), it is possible to obtain expressions for estimating both horizontal and vertical permeabilities, respectively:
3/ 2 1/ 2
3
max
12.3168 ( *
')
p w
q Bh r
k
h t
P
(16)2 max
41481.608
t zc h
k
t
(17)Notice that the second derivative is not used since the pressure derivative tendency is negative; then, the second derivative is going to be negative making impossible to plot on a log-log scale.
2.3. Conventional analysis
If the dimensionless quantities are replaced in Equation (6), it yields:
3/ 2 5/ 2 3/ 2 6278204.1 141.2 p t w sps z sps h c
q B r
P t k k q B s kh
(18)
Equation (18) suggests that a Cartesian plot of Pwf
versus t-3/2 for drawdown or of P
wf versus (tp+t) -3/2+t -3/2
will provide a straight line which slope, msps, and intercept, bsps, allow obtaining horizontal permeability (if kz is
known) and spherical stabilization skin, respectively;
1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
t , hr
P & t * P ', p s i
( *t P')sps0.188 psi
( *t P')r1.577 psi 0.14 hr spsri t 0.637 hr sps t
max 98.51 hr
t
6 max ( *t P') 7.4 10 psi
Radial flow regime horizontal line
Figure-4. Pressure and pressure derivative versus time
log-log plot for example 1.
3/ 2 5/ 2
6278204.1 w p t
z sps
h c q B r k k m
(19)
141.2
sps sps
khb
s
q B
(20)As observed here, conventional analysis has a strong limitation if the vertical is unknown. In such case the best result from conventional analysis is to find the k kz3/2:
5/ 2 3/2
3/ 2
6278204.1
w(
p t)
zsps
q
B r h c
kk
m
3. EXAMPLES
3.1. Synthetic example 1
A synthetic pressure test data was obtained with the information given below. Generated pressure drop and pressure derivative data are given in Figures-4 and 5, respectively.
B = 1.0 bbl/STB q= 200 STB/D
h = 200 ft = 1 cp
rw = 0.3 ft ct= 3x10-6 psi-1 Pi = 5000 psi = 10 % k = 50 md hp= 40 ft
CD = 0 s = 0
kz = 5 md
Estimate horizontal and vertical permeabilities.
Solution by TDS technique
The following information was read from Figure-4:
tmax = 98.51 hr (t*P’)max = 7.4x10-6 psi tsps = 0.637 hr (t*P’)sps = 0.188 psi
70 71 72
0 1 2 3 4 5 6 7 8 9 10
P
, p
s
i
wf m = -0.0615 p
si*hr3/2
t , -3/2 -3/2hr
Figure-5. Pressure versus time to the power -3/2 for
example 1.
Find horizontal permeability with Equation (16) using the maximum point pressure derivative:
3/ 2
3 6
200(1)(1)(40) 0.3
38 md 12.3168(200) (7.4 10 )
k
Estimate the vertical permeability with Equation (17) using the time at which the maximum derivative occurs:
6 2
41481.608(0.1)(1)(3 10 )(200)
5.05 md 98.51
z
k
Use any arbitrary point on the spherical stabilization straight line and find permeability with Equation (8):
5 / 2
3 / 2 6
9 4 1 7 3 0 6 . 1 5 4 ( 2 0 0 ) ( 1 ) (1 ) 0 . 3 0 . 1 8 8
( 4 0 ) ( 0 . 1 ) ( 3 1 0 )
3 9 . 5 m d ( 5 . 0 8 ) ( 0 . 6 3 7 )
k
Since permeability is known now, find with Equation (12) the value of the pressure derivative during radial flow regime masked by the effect of constant-pressure boundary:
70.6(200)(1)(1)
( * ') 1.7874 psi
(200)(39.5)
r
t P
Now, draw a horizontal line going through 1.7874 psi, see Figure-4, and read the intersection of that line with the spherical stabilization straight line, extrapolated if needed,
tspsrsi = 0.14 hr
1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
t , hr
P
&
t
*
P
',
ps
i
( *t P')sps2.784 psi
( *t P')r10.63 psi
0.03 hr spsri
t
0.1 hr sps
t
max 17.5 hr
t
5 max
( *t P') 8.73 10 psi
Radial flow regime horizontal line
Figure-6. Pressure and pressure derivative versus time
log-log plot for example 2.
Estimate again vertical permeability using Equation (13);
15 6 2
5.0627 (40)(0.1)(3 10 )(0.3)
5.06 psi 200(1.55)
z
k
Solution by conventional analysis
A slope value of -0.0615 psi*hr3/2 was read from
Figure-5. Assuming vertical permeability is known then horizontal permeability is estimated by means of Equation (19),
5 / 2
3 / 2 6
6 2 7 8 2 0 4 . 1 ( 2 0 0 ) ( 1 ) ( 1 ) 0 . 3
0 . 0 6 1 5
( 4 0 ) ( 0 . 1 ) ( 3 1 0 )
4 0 . 9 4 m d
5 k
3.2. Synthetic example 2
Another simulated example generated with the below information is reported in Figures-6 and 7. It is required to find the permeabilities from this test.
B = 1.15 bbl/STB q= 320 STB/D
h = 120 ft = 3 cp
rw = 0.4 ft ct= 5x10-5 psi-1 Pi = 3500 psi = 7 % k = 60 md hp= 30 ft
CD = 10 s = 0
kz = 72 md
Estimate horizontal and vertical permeabilities.
Solution by TDS technique
The following information was read from Figure-6:
tmax = 17.5 hr (t*P’)max = 8.73x10-5 psi tsps = 0.1 hr (t*P’)sps = 2.784 psi
250 260 270 280 290
0 100 200 300 400 500
P
,
ps
i
wf
m = -0.055 ps
i*hr3/2
t , -3/2 -3/2hr
Figure-7. Pressure versus time to the power -3/2 for
example 2.
As for the former example, Equation (16) is used to determine the horizontal permeability using the maximum point pressure derivative:
3/2
3 5
320(3)(1.15)(30)
0.4
61.75 md
12.3168(120) (8.73 10 )
k
Now, using Equation (17) find the vertical permeability:
5 2
41481.608(0.07)(3)(1 10 )(120)
71.7 md 17.5
z k
Verify the value of the horizontal permeability with Equation (8):
5/2
3/2 5
9417306.154(320)(3) (1.15) 0.4 2.784
(30)(0.07)(1 10 )
61.2 md (71.7)(0.1)
k
Any of the above horizontal permeability values, or an average, can be used to find (t*P’)r = 10.63 psi
from Equation (12). After drawing the horizontal line, the intersection point read is tspsrsi = 0.03 hr which allows
estimating again the vertical permeability using Equation (13);
5 1/3 2/3
2610.65(30)(0.07)(3)(1 10 )(0.4) (120) 0.03
72.4 md
z
z
k
k
Solution by conventional analysis
A slope value of -0.055 psi*hr3/2 was estimated
from Figure-7. Assuming vertical permeability is known then horizontal permeability is estimated using Equation (19),
Table-1. Summary of results from examples.
Example 1
Parameter Actual TDS Equation
number
k, md 50 38 16
k, md 50 39.5 8
kz, md 5 5.05 17
kz, md 5 5.06 13
Parameter Actual Conventional
analysis
Equation number
k, md 50 40.94 19
Example 2
Parameter Actual TDS Equation
number
k, md 60 61.75 16
k, md 60 61.2 8
kz, md 72 71.7 17
kz, md 72 72.4 13
Parameter Actual Conventional
analysis
Equation number
5/ 2
3/2 5
6278204.1(320)(3) (1.15) 0.4 0.055
(30)(0.07)(1 10 )
64.1 md 72
k
4. COMMENTS ON THE RESULTS
It is observed from Table-1 a good agreement between the actual and obtained results for both examples. Horizontal permeability estimated by either technique in the first example did not agree very much with the actual value, maybe, because it is a statistical model, but the vertical permeability results were excellent. In the second exercise both permeability provided a very good agreement with the actual data used for the simulation.
CONCLUSIONS
a) Expressions for the characterization of the spherical stabilization flow regime were introduced and satisfactorily tested with synthetic examples. This regime can help in estimation of vertical permeability of reservoir.
b) Spherical stabilization flow regime can prove very important for estimation of anisotropy ratio in analysis of pressure data obtained by wireline formation tester where pretest points are near constant pressure boundaries.
c) For the TDS technique two equations for estimating horizontal permeability and two equations for estimating vertical permeability were introduced and tested. The permeability estimation does not depend on each other. For the case of conventional analysis one of the two permeabilities must be known to find the other one. This demonstrates the advantage and power of the TDS technique.
ACKNOWLEDGEMENTS
The authors thank Universidad Surcolombiana, Universidad del Atlántico and the University of Oklahoma for providing financial support for the completion of this study.
Nomenclature
B Volume factor, rb/STB
B Partial penetration ratio = hp/h C Wellbore storage coefficient, bbl/psi
ct Total system compressibility, psi-1 H Reservoir thickness, ft
hp Perforated interval, ft
Iv Vertical anisotropy, vertical to horizontal
permeability ratio
K Reservoir horizontal permeability, md
kz Reservoir vertical permeability, md msps Slope of the P. vs. t-3/2 plot
P Pressure, psi
Pi Initial reservoir pressure, psi Pwf Wellbore flowing pressure, psi
Q Water flow rate, BPD
rw Wellbore radius , ft S Skin factor
T Time, days
tD Dimensionless time coordinate tD*PD’ Dimensionless pressure derivative
(t*P’) Pressure derivative
Greeks
Porosity, fraction
Viscosity, cp
Suffices
D Dimensionless
i Initial
max Maximum before steady-state regime
develops
r Radial
sps Spherical stabilization
spsri Intercept of spherical stabilization and radial
straight lines
wf Well flowing
ws Well static
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